00:01
Okay, so your density function is f of x equals 1 over 2 times e to negative 1 over 2 times x.
00:09
So this basically follows the exponential function, or excuse me, exponential random variable with rate parameter 1 over 2.
00:18
So i'm going to solve the expected value and the variance in the general case.
00:27
Case.
00:28
So here i describe the or depict the rate parameter as lambda.
00:39
So first we'll find the expected value and the domain is x greater than 0.
00:47
So we take the integral from 0 to infinity x times the pdf x e to negative lambda x dx.
01:05
Okay now we're going to use the integration by parts.
01:09
So we're going to let f equals x, then f prime equals 1, and g prime equals e to negative lambda x, then g equals negative 1 over lambda e to negative lambda x.
01:29
Okay, so the integral becomes lambda times negative x e to negative lambda x over lambda from 0 to infinite plus 1 over lambda integral e to negative lambda x dx 0 to infinite.
02:00
Okay, now we can plug out 1 over lambda and obtain negative x e to negative lambda x, 0 to infinite, plus we can just integrate this 1 over lambda e to negative lambda x from 0 to infinite.
02:20
It.
02:22
And so the first one is evaluates to 0, negative 1 over lambda, 0 minus 1.
02:30
So we get the expected value as 1 over lambda.
02:34
And in your case, since lambda equals 1 over 2, we have 1 over 1 over 2, or 2...